# NCERT Solutions for Class 12 Maths Chapter 11 – Three Dimensional Geometry

NCERT Solutions for Class 12 Maths Chapter 11- Three Dimensional Geometry cover all significant concepts present in this chapter. Some of them are the central idea of 3D, axes, the direction of the cosine lines, coordinate planes, skew lines, equation of a vector of a line, Cartesian equation of a line passing through two points, the centroid of a tetrahedron, Direction ratios, the distance of point P from coordinate axes, relation between two lines and the angle between two planes.

CBSE NCERT Class 12 Maths Chapter 11 consists of 3 exercises and a miscellaneous exercise that give you ample practice of the various question styles that are important from board exam perspective. Our NCERT Solutions answer all questions from all exercises in detail to help you understand quickly and learn quickly.

They help you develop an in-depth understanding of all mathematical concepts of this chapter and its various applications. They help you delve beyond the fundamental concepts and dive deeper than the advanced concepts covered in this chapter to help you become thorough with the chapter and develop a strong conceptual base. Our Maths shortcuts and effective study strategies help you increase your overall speed and efficiency. They are just what you need to excel in both your Board exams and various competitive exams.

**NCERT CBSE Class 12 Maths Chapter 11**

The chapter Three Dimensional Geometry is one of the crucial chapters of Class 12 Maths Syllabus. It covers various significant topics and subtopics. Some of them have been mentioned below.

**Central Idea of 3D**

This section tells you that there is an infinite number of points in space. We have to attempt to identify every point in space through the use of three mutually perpendicular coordinates axes OX, OY, and OZ.

**Axes**

Here you will learn that the three axes are the three mutually perpendicular coordinates OX, OY and OZ.

**Coordinate Planes**

This section of NCERT Class 12 Maths Chapter 11 tells you that coordinate planes that can be formed with the help of x and y-axis are called x – y planes, similarly, planes formed with z and x-axis are called z – x planes.

**The direction of the Cosine Line**

Under this section of the CBSE NCERT Class 12 Maths Book, you will study the concepts of direct cosine of line. Here you will learn about cosine angles subtended bylines on the positive coordinate axes.

**Skew lines**

In this topic, NCERT Solutions for Class 12 Maths Chapter 11- Three Dimensional Geometry, you will get to know about skew lines that exist in different planes, but neither intersect nor go parallel. From this chapter, you will also learn the methods of finding the angle between two intersecting lines that are parallel to one another.

**Equation of the vector of a line**

We discuss in this section, the method of finding out the vector equation of a line passing through a point denoted by position vector. It is one of the most vital modules of NCERT CBSE Class 12 Maths Chapter 11.

### Cartesian equation of a line passing through two points

In this topic, you will know about the equation for solving cartesian equation three- dimensional geometry.

If there are two points with dimensions (x_{1}, y_{1}, z_{1}, ) and (x_{2}, y_{2}, z_{2}, )

Then x−x_{1}, /x_{2}, − x_{1} =y−y_{1}, /y_{2}, −y_{1} =z−z_{1}, /z_{2}, −z_{1}

Some other significant topics covered in NCERT Class, 12 Maths Chapter 11 in details, include the angle between two planes, equation of a plane, projection of a line segment on a line, Direction ratios, relation between two lines, the centroid of a tetrahedron, section formula, coordinates of a point, the distance of point P from coordinate axes, and vector representation of a point in space.

## NCERT Solutions for Class 12 Maths Chapter 11-Three Dimensional Geometry Exercises

NCERT for Class 12 Maths Chapter 11- Three Dimensional Geometry consists of answers to all 3 exercises of this unit. These exercises are all-encompassing and help you become thorough with all concepts and theorems covered in the chapter. The components of the exercises have been discussed below.

**Exercise 11.1**

Exercise 11.1 has a total of 5 questions. This exercise will help you in understanding the theorems of the Cartesian coordinate system. All the questions deal with different facets of direction cosines, direction ratios.

**Exercise 11.2**

Under this exercise, you will come across 17 questions, all of which are long answer types. The questions follow the pattern of CBSE board examination papers and will help you in understanding the kind of questions that are common from this module. In the first few questions, you will find questions where we ask to find out whether the lines with given direction cosines are perpendicular or parallel to one another. Question 4 deals with vector representation of a point in space. Question number 6, needs you to find out the Cartesian equation of the lines passing through a given point and parallel to another line represented by an equation. Questions 7, 8 and 9 deal with the Cartesian equation of a line passing through two points, and vector equations. Questions 14 and 15 ask you to find the shortest distance between the given lines. Questions 16 and 17 also ask you to find the shortest distance between the lines whose vector representations are given.

**Exercise 11.3**

In this exercise of the NCERT Solutions for Class 12 Maths Chapter 11- Three Dimensional Geometry,** **there are a total of 13 questions. Here you will find both long and short answer type questions. The questions in this exercise cover various topics such as with the direction of cosine, vector equation, coordinates of parallel and perpendicular, and the Cartesian equation we ask in this exercise.

This chapter also comprises a miscellaneous exercise that contains 23 questions on different topics of three-dimensional geometry.

## Benefits of CBSE NCERT Solutions for Class 12 Maths Chapter 11: Important Topics

There are many benefits of learning and revising from our smart NCERT Solutions for Class 12 Maths Chapter 11- Three Dimensional Geometry. Some of them have been cited below.

- They are easy to understand, fully accurate and step by step solutions that help you learn quickly and effectively.
- Our NCERT Solutions include a wide variety of questions that help prepare during the examination.
- This compact study material makes learning complex theorems fun and interesting.
- They have been solved using the best methods of solving the problems.
- They have been designed to clear all your doubts and questions and improve your overall retention rate.
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